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Magazine Chapter 1: Problem 18
Find all complex numbers \(z\) such that $$ 4 z^{2}+8|z|^{2}=8 $$
Short Answer
Expert verified
Based on the step-by-step solution provided, the complex numbers z can be represented as:
$$
z = a \pm b i
$$
where \(a\) and \(b\) satisfy the conditions: \(1 \geq a^2 \geq \frac{2}{3}\) and \(b = \pm \sqrt{2 - a^2}\).
Step by step solution
01
Express z as a sum of real and imaginary parts
Let's express the complex number \(z\) as a sum of its real and imaginary parts, i.e. \(z = a + bi\).
02
Substitute z into the equation
Now, we will substitute the expression \(z = a + bi\) into the given equation and simplify:
$$
4(a+bi)^2 + 8|(a+bi)|^2 = 8
$$
03
Expand and simplify
First, expand \((a+bi)^2 = (a^2-b^2)+2abi\). Next, recall that \(|z|^2 = a^2+b^2\), so we have:
$$
4\left[(a^2-b^2)+2abi\right] + 8(a^2 + b^2) = 8
$$
Now, dropping the imaginary part, and keeping the real part of the equation:
$$
4(a^2-b^2) + 8(a^2 + b^2) = 8
$$
04
Simplify the equation further
Combine the terms on the left side of the equation:
$$
12a^2 + 4b^2 = 8
$$
05
Divide by the common factor
Divide by the common factor \(4\) to simplify even further:
$$
3a^2 + b^2 = 2
$$
06
Express b^2 in terms of a^2
Express \(b^2\) in terms of \(a^2\):
$$
b^2 = 2 - 3a^2
$$
07
Find the values of a and b
Since the right-hand side of the equation is a difference, there are two scenarios: (1) \(a\) is positive or zero, and (2) \(a\) is negative.
(1) If \(a\geq 0\), then we know that \(1 \geq 3a^2 \geq 0\). So, we get:
$$
0 \leq 3a^2 \leq 2 \Rightarrow 1 \geq a^2 \geq \frac{2}{3}
$$
And for each possible value of \(a\), we find the corresponding value of \(b\). Since \(a^2+b^2=2\), we can write:
$$
b = \pm \sqrt{2 - a^2}
$$
(2) If \(a<0\), then \(1 \geq 3a^2 \geq 0\). So, we obtain:
$$
0 \leq 3a^2 \leq 2 \Rightarrow 1 \geq a^2 \geq \frac{2}{3}
$$
And for each possible value of \(a\), we find the corresponding value of \(b\). Since \(a^2+b^2=2\), we can write:
$$
b = \pm \sqrt{2 - a^2}
$$
08
Combine the values of a and b to find the complex numbers z
Now, we have all the possible values of \(a\) and \(b\) that satisfy the given equation. Hence, we find the complex numbers \(z = a + bi\) in terms of the possible values of \(a\) and \(b\). Our solution set for \(z\) will be:
$$
z = a \pm b i
$$
where \(a\) and \(b\) satisfy the conditions we found above: \(1 \geq a^2 \geq \frac{2}{3}\) and \(b = \pm \sqrt{2 - a^2}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Real and Imaginary Parts
Complex numbers are numbers that have both a real part and an imaginary part. The real part is a number without an imaginary unit, represented as "a" in a complex number. The imaginary part appears with an "i," which stands for the square root of -1, and is represented as "b" in a complex number. Therefore, any complex number can be expressed as \(z = a + bi\).
- The real part of \(z\) is \(a\).
- The imaginary part of \(z\) is \(bi\).
Modulus of a Complex Number
The modulus of a complex number \(z = a + bi\) is a measure of its distance from the origin in the complex plane. It is calculated as \(\sqrt{a^2 + b^2}\). The concept is similar to finding the magnitude of a vector.
- The modulus gives a sense of the "size" of the complex number.
- It helps indicate how far a number is from zero in the complex plane.
Solving Quadratic Equations
Quadratic equations can be solved with complex numbers when they do not have real solutions. The general form of a quadratic equation is \(ax^2 + bx + c = 0\). If the discriminant \(b^2 - 4ac\) is negative, the equation has no real roots, and complex solutions are present.
- Solutions are found using the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
- For negative discriminants, \(\sqrt{b^2 - 4ac}\) involves imaginary numbers.
Complex Plane
The complex plane is a geometric representation of complex numbers, where each number corresponds to a point. It consists of a horizontal axis for the real part and a vertical axis for the imaginary part. This visual system is akin to a coordinate system for complex numbers.
- The real part of a complex number corresponds to the x-axis.
- The imaginary part corresponds to the y-axis.
Algebraic Representation of Complex Numbers
Algebraic representation provides a fundamental way to handle complex numbers mathematically. Using the form \(z = a + bi\), algebraic operations like addition, subtraction, multiplication, and division are conducted.
- To add or subtract, combine like terms: the real parts and the imaginary parts separately.
- To multiply, use distributive laws and remember that \(i^2 = -1\).
- Division involves multiplying by the conjugate to remove imaginary units from the denominator.
